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IMO #1 1959 (Hardest IMO question ever)

Pose Olympiad-level questions to the monkeys in the peanut #.
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IMO #1 1959 (Hardest IMO question ever)

Postby stupidityismygam » Wed Apr 18, 2007 11:34 pm

Prove that the fraction [unparseable or potentially dangerous latex formula] is irreducible for every natural number [unparseable or potentially dangerous latex formula]
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Re: IMO #1 1959 (Hardest IMO question ever)

Postby makashi » Thu Apr 19, 2007 2:25 am

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Postby Dragon of Light » Thu Apr 19, 2007 3:50 pm

How did you come up with that solution....

I never figure out what motivates these things...
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Postby stupidityismygam » Thu Apr 19, 2007 3:56 pm

well if it is irreducible then the gcd=1

if the gcd = 1, then there is a well known theorem that states:

for positive integers a,b

they are two integers x,y such that

[unparseable or potentially dangerous latex formula]
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Postby Dragon of Light » Thu Apr 19, 2007 4:08 pm

Darn, you're too smart. I quit.
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Postby stupidityismygam » Thu Apr 19, 2007 4:10 pm

you are so right...


you know, since you mae MOP, and i dont even know what the AIME is...
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Postby makashi » Thu Apr 19, 2007 5:34 pm

Actually, I just remembered the solution from when I first tried and failed it.
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Postby stupidityismygam » Thu Apr 19, 2007 5:37 pm

cheater :)
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Postby kashhustler » Thu Apr 19, 2007 10:22 pm

By the Euclidean Algorithm, GCD(21n+4, 14n+3) = GCD(14n+3, 7n+1) = GCD(7n+2, 7n+1) = GCD(7n+1, 1) which is obviously 1 so 21n+4 and 14n+3 are relatively prime
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